Chapter 10

Complex Numbers

Chapter description
In this final chapter we powerfully extend the possibilities of what we can do with our number system. This extension leads us to simpler ways of finding mathematical rules. It also has many very important physical applications.
The chapter is divided into the following sections.

  1. A new sort of number
    1. Finding the missing roots
    2. Finding roots for all quadratic equations
    3. Modulus and Argument (or mod and arg for short)

  2. Doing arithmetic with complex numbers
    1. Addition and subtraction
    2. Multiplication of complex numbers
    3. Dividing complex numbers in mod arg form
    4. What are complex conjugates?
    5. Using complex conjugates to simplify fractions

  3. How e connects with complex numbers
    1. Two for the price of one -- equating real and imaginary parts
    2. How does e get involved?
    3. What is the geometrical meaning of e jt ?
    4. What is e-jt and what does it do geometrically?
    5. A summary of the sin/cos and sinh/cosh links
    6. De Moivre's theorem
    7. Another example: writing cos 5x in terms of cos x
    8. More examples of writing trig functions in different forms
    9. Solving a differential equation which describes SHM
    10. A first look at how we can use complex numbers to describe electric circuits

  4. Using complex numbers to solve more equations
    1. Finding the n roots of zn =a + bj
    2. Solving quadratic equations with complex coefficients
    3. Solving cubic and quartic equations with complex roots

  5. Finding where z can be if it must fit particular rules
    1. Some simple examples of paths or regions where z must lie
    2. What do we do if z has been shifted?
    3. Using algebra to find where z can be
    4. Another example involving a relationship between w and z
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